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Statistical Mechanics Derived from Quantum Mechanics

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Statistical Mechanics Derived from Quantum Mechanics Statistical Mechanics Derived From Quantum Mechanics Yu-Lei Feng1 and Yi-Xin Chen1 1Zhejiang Institute of Modern Physics, Zhejiang University, Hangzhou 310027, China [email protected] [email protected] Abstract A pedagogical derivation of statistical mechanics from quantum mechanics is pro- vided, by means of open quantum systems. Besides, a new definition of Boltzmann entropy for a quantum closed system is also given to count microstates in a way con- sistent with the superposition principle. In particular, this new Boltzmann entropy is a constant that depends only on the dimension of the system's relevant Hilbert subspace. Finally, thermodynamics for quantum systems is investigated formally. arXiv:1501.05402v2 [hep-th] 7 Apr 2015 Contents 1 Introduction 1 2 A Brief Introduction to Entanglement Entropy 3 3 From Quantum Mechanics to Statistical Mechanics 5 3.1 Statistical mechanics derived from quantum mechanics . .5 3.2 Boltzmann entropy for a quantum closed system . 12 3.3 Count microstates for a single quantum free particle . 18 3.4 Thermodynamics and statistical mechanics for quantum systems . 20 4 Conclusions 27 A Statistical mechanics for an ideal quantum gas 27 1 Introduction According to the number of degrees of freedom, a system is usually classified into three cases: microscopic, macroscopic and cosmoscopic. It's believed that the corresponding descriptions for those three cases are also different. For example, for a simple microscopic system quantum mechanics is enough; while for the latter two, such as a complex macro- scopic system, (quantum) statistical mechanics is more useful. In (quantum) statistical mechanics [1], Boltzmann entropy SBolt is an important quantity that provides a measure of the number of microstates in a macrostate denoted by macro-quantities. Besides, ther- modynamics can be obtained by means of ensemble theory, microcanonical ensemble for isolated or closed systems, (grand) canonical ensemble for open systems. A macrostate for a macroscopic system is usually denoted as (E; SBolt; N; V ), given by its (internal) energy E, Boltzmann entropy SBolt, particle number N, space volume V etc. These macro-quantities can be treated as observables of the studied system, and are usually obtained through measurements. In particular in quantum statistical mechanics, quantum observables H^ , N^ can be assigned so that their corresponding macro-quantities ^ can be obtained by taking averages in terms of T r(^ρstO), withρ ^st the density matrix for 1 one of those familiar ensembles . Moreover, the Boltzmann entropy SBolt can be expressed formally (not determined) by the von Neumann entropy S[^ρst]. 1The space volume V is an exception, indicating that it may be a classical quantity, just like its corresponding \force", pressure p. 1 More specifically, the density matrixρ ^st in quantum statistical mechanics can be given by a general form [1] N 1 X ρ^ (t) = j k(t)ih k(t)j; (1) st N k=1 where the sum is over the N member systems of a presumed ensemble. The state j k(t)i is some normalized quantum state for the member system k at time t. For different ensem- bles,ρ ^st will be reduced to some corresponding forms. For example, for microcanonical ensemble, according to the postulate of equal a priori probabilities, we will have the matrix elements (^ρst)mn = cδmn [1] in some representation. Easily to see, the \presumed ensem- ble", composed of huge number of \mental copies" of the studied system, is an independent ^ concept. This can be seen by noting that, in the expression T r(^ρstO) there are two kinds of averages, an ensemble average and a quantum average. As a result, when calculating PN k the von Neumann entropy S[^ρst], the presupposed probability distribution k=1 P = 1 for the presumed ensemble will lead to some extra fictitious information, in addition to the quantum information encoded in the system's quantum states. The fictitious information is actually resulted from our uncertainty about the studied system. But this uncertainty is only artificial, not intrinsical for the system, especially for quantum systems. For example, a quantum closed system can be well described by a unitary evolution from some given initial state. However in statistical mechanics, microcanonical ensemble will be applied to describe a closed system. Then some fictitious information (about the uncertainty of the initial states) may be added, destroying the quantum coherence of the closed system. In a word, the Boltzmann entropy given by quantum statistical mechanics seems not to be exact, due to the implicit fictitious information of the presumed ensemble. Certainly, the above fictitious information is not serious for ordinary thermodynamics, in which the Boltzmann entropy is only a coarse grained quantity with additivity. This is not the case for a quantum mechanics description, in which von Neumann entropy is a fine grained quantity with only subadditivity [2, 3]. In fact, thermodynamics can be treated as an effective description of the underlying quantum details. Besides, theρ ^st used in those familiar ensembles are actually determined according to some (extreme) conditions of the presumed ensemble. While the density matrix in von Neumann entropy must be operated always according to the quantum rules. Therefore, to obtain some more correct description, we should restrict ourselves always in the framework of quantum mechanics. In this paper, we try to provide a derivation of statistical mechanics from quantum me- chanics, so that the fictitious information from the concept of ensemble can be removed. We show that the (grand) canonical ensemble theory of statistical mechanics can be de- 2 rived effectively by means of open quantum systems. In particular, the most probable distribution for statistical mechanics will correspond to some stable one for the studied open quantum system. In this way, the Boltzmann entropy for the (grand) canonical en- semble can be treated as an approximation of some entanglement entropy in the stable limit. Furthermore, a new definition of Boltzmann entropy for a quantum closed system is also given, which counts microstates in a way consistent with the superposition principle of quantum mechanics. In particular, that new Boltzmann entropy is a constant that doesn't depend on the system's energy and space volume. In fact, it is identical to the maximum von Neumann's entropy related to the density matrixes for some measurements, thus it depends only on the dimension of the system's relevant Hilbert subspace. This paper is organized as follows. In Sec. 2, some basic properties of entanglement entropy or von Neumann entropy are briefly mentioned for comparing to the Boltzmann entropy. In Sec. 3.1, some general investigations are given to show that the (grand) canon- ical ensemble theory of statistical mechanics can be derived effectively by means of open quantum systems. While in Sec. 3.2, a new Boltzmann entropy is defined for a quantum closed system, which is a constant independent of the system's energy and space volume. Some more details for thermodynamics of quantum systems, especially the thermal equi- librium between two macroscopic subsystems are given in Sec. 3.4. In addition, two simple examples are also analyzed in Sec. 3.3 and the Appendix A. 2 A Brief Introduction to Entanglement Entropy In this intermediate section, a brief introduction to entanglement entropy is provided to compare with the Boltzmann entropy. An entanglement entropy is defined as Sen = −T r(^ρext lnρ ^ext); (2) which is constructed from some reduced density matrix. A reduced density matrix could occur only if the relevant system contained two or more than two independent degrees of freedom2. According to the standard results of quantum mechanics, a complete collection of observables can be expressed generally as ^ ^ ^ ^ fO1; O2; · · · g; [O1; O2] = ··· = 0; (3) 2Specially, when making a measurement on a single quantum degree of freedom, the apparatus should also be included to form a larger closed system. 3 with \··· " denoted as the observables relevant for the rest degrees of freedom. Then the full quantum state can be expressed by X j i = C1;2;···jO1;O2; · · · i; (4) 1;2;··· from which some reduced density matrix can be derived, for exampleρ ^1 = T r2;···j ih j. There is not a necessary second law for entanglement entropy in the microscopic sense. For example, consider two qubits undergoing the following two unitary processes (αj0ia + βj1ia)j0ib ! αj0iaj0ib + βj1iaj1ib ! (αj0ia + βj1ia)j0ib; (5) i.e. undergoing two C-NOT gates [2, 3]. We can calculate the reduced density matrix ρ^a and its corresponding entanglement entropy Sa. Easily to see, for the initial and final cases, Sa = 0, while for the intermediate case, Sa 6= 0. It is believed that the first process in Eq. (5) gives a second law for entanglement entropy, since the correlation is generated through the interaction [2, 3]. However, for simple quantum systems the interactions which can decouple (effectively) the already correlated systems, such as the one for the second process in Eq. (5), also occur frequently in the microscopic sense. Therefore, no second law is necessary for entanglement entropy that can always be calculated for simple quan- tum systems undergoing various basic or microscopic evolutions. However, for complex macroscopic systems, the involved interactions are so complicated that the decoupling or decorrelation is usually hard in the macroscopic sense. In this case, a Boltzmann entropy seems to be more useful, since entanglement entropy is difficult to be calculated due to the complexity. This will be discussed in the next section. Since entanglement entropy or von Neumann entropy is wildly used in quantum in- formation theory, its meaning is similar to the (classical) Shannon entropy. Generally speaking, it's a measure of the uncertainty before we learn a system, or a measure of how much information we have gained after we learn that system [2, 3].
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